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  • Abstract

Spin-Wave Instabilities in Spin-Transfer-Driven Magnetization Dynamics

We study the stability of magnetization precessions induced in spin-transfer devices by the injection of spin-polarized electric currents. Instability conditions are derived by introducing a generalized, far-from-equilibrium interpretation of spin waves. It is shown that instabilities are generated by distinct groups of magnetostatically coupled spin waves. Stability diagrams are constructed as a function of external magnetic field and injected spin-polarized current. These diagrams show that the application of larger fields and currents has a stabilizing effect on magnetization precessions. Analytical results are compared with numerical simulations of spin-transfer-driven magnetization dynamics.

SECTION I

INTRODUCTION

Currents of spin-polarized electrons can induce large-amplitude magnetization precessions at microwave frequencies in small-enough magnetic devices [Slonczewski 1996, Berger 1996]. There is mounting experimental evidence that these so-called spin-transfer phenomena do occur in nanopillar or nanocontact devices under current densities of the order of 106−108 A/cm2 [Kiselev 2003, Rippard 2004, Krivorotov 2007, Boone 2009]. This discovery has boosted the already widespread interest in the physics of the interplay between magnetism and electron transport, and has triggered efforts toward the promising development of new generations of microwave spin-transfer nanooscillators.

A spin-transfer device is a nonlinear open system, driven far-from-equilibrium by the action of the spin-polarized electric current. The excited magnetization precessions represent strong excitations of the magnetic medium, which, in principle, may give rise to various types of instability and eventually to transitions to chaotic dynamics [Ji 2003, Polianski 2004, Zhu 2004, Lee 2004, Slavin 2008]. A parallel can be drawn with ferromagnetic-resonance Suhl’s instabilities [Suhl 1957] in which certain spin waves can get coupled to the uniform precession and start to grow to large nonthermal amplitudes, thus destroying the spatial uniformity of the original state.

In this letter, we demonstrate that spin-wave instabilities may occur in spin-transfer-driven magnetization dynamics as well. However, the system is far from equilibrium and the classical notion of spin waves fails. Indeed, it is the large-amplitude magnetization precession induced by spin transfer that plays the role of reference state, and spin waves only exist in a generalized, nonequilibrium sense, as small-amplitude perturbations of this state [Bertotti 2001, Kashuba 2006, Garanin 2009]. This scenario emerges with clarity in the time-dependent vector basis in which the reference magnetization precession is stationary. The spin-wave equations in this basis are characterized by two features: 1) a well-defined dispersion relation ω(q; cos θ0), whose nonequilibrium nature is revealed by its explicit dependence on the magnetization precession amplitude cosθ0; and 2) the presence of time-periodic coupling terms due to the magnetostatic fields generated by individual spin waves. This coupling leads to the appearance of narrow instability tongues around the parametric resonance condition ω (q; cos θ0) ∼ ω0, where ω0 is the magnetization precession angular frequency.

A preliminary, mostly mathematical analysis of the problem was proposed in [Bertotti 2009]. In this letter, we show that spin-wave instabilities occur only for particular combinations of external magnetic field and injected spin-polarized current. In addition, instabilities result in limited spatial and temporal distortions, which somewhat obscure, but yet do not completely disrupt the precessional character of the original state. This robustness of excited precessions with respect to spin-wave instabilities has a precise physical origin. Indeed, the discrete nature of the spin-wave spectrum caused by boundary conditions in submicrometer devices reduces the number of available spin-wave modes, which can contribute to instabilities. On the other hand, the strength of the magnetostatic effects responsible for instabilities is drastically reduced, due to the ultrathin nature of spin-transfer devices, and instability thresholds are consequently enhanced. Finally, spin-transfer-driven precessions are characterized by large amplitudes, and as such, are less easily masked by the onset of nonuniform modes. In spin-transfer nanooscillators, spin-wave instabilities are expected to result in increased oscillator linewidths, a conclusion that might explain some of the puzzling experimental results obtained in this area [Mistral 2006].

SECTION II

PHYSICAL MODEL

To start the technical discussion, consider an ultrathin disk with negligible crystal anisotropy (e.g., permalloy). Typically, this disk will be the so-called free layer of a nanopillar spin-transfer device (see inset in Fig. 1). The disk plane is parallel to the (x,y) plane and is traversed by a flow of electrons with spin polarization along the ez direction. The dimensionless equation for the dynamics of the normalized magnetization m(r,t) (|m|2 = 1) in the disk, in the presence of spin transfer and neglecting the Oersted field due to the electric current, is as follows [Slonczewski 1996, Bertotti 2005]:Formula TeX Source $$\eqalignno{{\partial {\bf m} \over \partial t} &- \alpha \, {\bf m}\times { \partial {\bf m}\over \partial t} = - {\bf m}\times ({\rm h}_{az}{\bf e}_{z}+ {\bf h}_{M}+ \nabla^2 {\bf m}- \beta \, {\bf m}\times {\bf e}_{z}).\qquad&\hbox{(1)}}$$Here, the external magnetic field hazez and the magnetostatic field hM are measured in units of the spontaneous magnetization Ms, time in units of (γ Ms)−1 (γ is the absolute value of the gyromagnetic ratio), and lengths in units of the exchange length. The field hM is the solution to magnetostatic Maxwell equations ∇ × hM = 0, ∇ ·hM = −∇ ·m with appropriate interface conditions. The external field is perpendicular to the disk plane, while the spin-transfer torque is simply proportional to the sine of the angle between m and ez. The parameter β is proportional to the spin-polarized current density (see [Bertotti 2005] for the detailed definition), and in typical situations, it is comparable with the damping constant α.

Figure 1
Fig. 1. Stability diagram in (haz, β/α) control plane for an ultrathin permalloy disk. System parameters: α = 0.02, d = 0.6, R = 23.6, and N = 0.02 (lengths are measured in units of the exchange length lEX = 5.72 nm). Magnetization is parallel to spin-polarization in region P; antiparallel to spin polarization in region A; precessing around the spin-polarization axis in regions O and SW. Dashed line is an example of line of constant precession amplitude (cos θ0 = 0.5) computed from (2). Spin-wave instabilities occur in region SW. Small framed area is shown in detail in Fig. 2. (Inset) Typical geometry of a nanopillar spin-transfer device.

Whenever |haz− β/α | ≤ NzN (Nz and N are the disk demagnetizing factors, with Nz+ 2 N = 1), (1) admits time-harmonic solutions m0(t), corresponding to spatially uniform precession of the magnetization around the z-axis [Bazaliy 2004] (see Fig. 1). The precession amplitude and angular frequency are as follows:Formula TeX Source $$\cos \theta_0 = { {\rm h}_{az}- \beta /\alpha \over N_{z}- N_{\bot }}, \omega_0 = { \beta \over \alpha }.\eqno{\hbox{(2)}}$$

To study the stability of m0(t), consider the perturbed motion m(r, t) = m0(t) + δm(r, t), with | δm(r, t) | ≪ 1. The corresponding magnetostatic field will be: hM(r, t) = − Nzm0 zNm0 ⊥+ δhM(r, t), where δhM represents the magnetostatic field generated by δm. Since, we are interested in ultrathin layers, we shall assume that δm does not depend on z: δm(r,t) = δm(x,y,t).

The perturbation δm is orthogonal to m0(t) at all times, since the local magnetization magnitude |m|2 = 1 must be preserved. Hence, it is natural to represent δm in the time-dependent vector basis (e1(t),e2(t) ) defined in the plane perpendicular to m0(t), with e2(t) parallel to ez×m0(t) and e1(t), such that (e1,e2,m0) form a right-handed orthonormal basis. The perturbation can be written as: δm(r, t) = δ m1(r, t) e1(t) + δ m2(r, t) e2(t). By linearizing (1) around m0(t) and averaging the linearized equation over the layer thickness, one obtains the following coupled differential equations in matrix form:Formula TeX Source $$\eqalignno{\left(\!\matrix{1 & \alpha\cr - \alpha & 1}\!\right) {\partial \over \partial t} \left(\!\matrix{\delta m_{1}\cr \delta m_{2}}\!\right) &= \left(\!\matrix{0 & 1\cr -1 & 0}\!\right) \left(\!\matrix{\langle \delta {\rm h}_{M}\rangle_1\cr\langle \delta {\rm h}_{M}\rangle_2} \!\right) \cr&\quad + \left(\!\matrix{0 & N_{\bot }+ \nabla^{2}_{\bot }\cr - N_{\bot }- \nabla^{2}_{\bot} & 0}\!\right) \left(\!\matrix{\delta m_{1}\cr \delta m_{2}}\!\right)\qquad&\hbox{(3)}}$$where ∇2 = ∂2/∂ x2 + ∂2/∂ y2, while < … > represents the z average over the thickness of the disk, and < δhM>1 = < δhM> ·e1(t), < δhM>2 = < δhM> ·e2(t).

To grasp the physical consequences of (3), consider the plane-wave perturbation δm(r, t) = a(t) exp (i q·r) in an infinite layer (N = 0). The corresponding magnetostatic field is as follows [Bertotti 2009]:Formula TeX Source $$\langle \delta {\bf h}_{M}\rangle = - s_q \, \delta {\bf m}_z - \left(1 - s_{q} \right) \delta {\bf m}_q; s_{q} = { 1 - \exp (-q d)\over q d}\eqno{\hbox{(4)}}$$where δmz = (δm·ez)ez and δmq = (δm·eq)eq, eq being the unit vector in the q direction. The field − sq δmz is generated by the magnetic charges at the layer surface, whereas the field − (1 − sq) δmq is due to volume charges. By taking into account that ∇2 δm = − q2 δm, δmz ·e1(t) = δ m1 sin2 θ0, and δmz ·e2(t) = 0, one finds from (3) that m0(t) is always stable with respect to the action of exchange forces and surface magnetic charges. Only volume charges can make the precession unstable. This conclusion follows from the fact that the z-axis, along which the surface-charge magnetostatic field is directed, is a symmetry axis for the problem. Surface-charge-driven instabilities may appear in nonuniaxial systems.

The 2-D and uniaxial character of the problem makes it natural to introduce polar coordinates (r, ϕ) in the disk plane, with the origin at the center of the disk. The natural boundary condition in polar coordinates is ∂δm/∂ r |r = R = 0, where R is the disk radius. The generic perturbation satisfying this boundary condition consists of cylindrical spin waves of the typeFormula TeX Source $$\delta {\bf m}(r, \phi, t) = \sum_{n = -\infty }^{+\infty } \sum_{k = 0}^{\infty } {\bf {a}}_{n k} (t) \, J_n \left(q_{n k} r \right) \, \exp (i n \phi)\eqno{\hbox{(5)}}$$where Jn (z) is the nth order Bessel function. The wave-vector amplitude qnk is identified by two subscripts because, for each n, it must satisfy the boundary condition ∂Jn (z)/∂ z = 0 for z = qnk R, which has infinite solutions qn 0, qn1, qn2, … of increasing amplitude. The cylindrical spin waves Fnk(r, ϕ) = Jn (qnk r ) exp (in ϕ) are a complete orthogonal set of eigenfunctions of the ∇2 operator: ∇2 Fnk(r, ϕ) = − qnk2 Fnk(r, ϕ). The magnetostatic field δhM can be computed by applying (4) to the plane wave integral representation: Fnk(r, ϕ) = 1/(2 π in) ∫02 πexp (i qnk·r) exp (in ψ) d ψ, where the polar representation of r and qnk is r = (r, ϕ) and qnk = (qnk, ψ), respectively. By following these steps, writing ank(t) as ank(t) = cnk,1(t) e1(t) + cnk,2(t) e2(t), and neglecting small terms proportional to N, (3) is transformed into the following system of coupled equations:Formula TeX Source $$\eqalignno{{ d c_{n k}\over d t} &= A_{n k} \, c_{n k} + \exp \left(2 i \omega_0 t \right) \,{\cal R} \,\sum_{p = 0}^{\infty } \, \Delta_{n k; p}^{+} \, \epsilon_{n+2,p} \, c_{n+2,p}\cr&\quad + \exp \left(- 2 i \omega_0 t \right) \,{\cal R}^* \,\sum_{p = 0}^{\infty } \, \Delta_{n k; p}^{-} \, \epsilon_{n-2,p} \, c_{n-2,p}&\hbox{(6)}}$$whereFormula TeX Source $$\eqalignno{c_{n k} &\equiv \left(\matrix{ c_{nk,1}\cr c_{nk,2}}\right)&{\hbox{(7)}}\cr A_{n k} &= { 1\over 1 + \alpha^{2}} \left(\matrix{ 1 & - \alpha\cr \alpha & 1}\right) \left(\matrix{ 0 & - \nu_{n k}\cr \nu_{n k} - \kappa_{n k} \sin^{2} \theta_{0} & 0}\right)&{\hbox{(8)}}\cr{\cal R} &= { 1\over 1 + \alpha^{2}} \, \left(\matrix{ 1 & - \alpha\cr \alpha & 1}\right) \left(\matrix{ i \, \cos \theta_{0} & -1\cr - \cos^{2} \theta_{0} & - i \, \cos \theta_{0}}\right)&{\hbox{(9)}}\cr\Delta_{n k; p}^{\pm } &={ 1\over \Delta_{n k}} \int_{0}^{R} r \, J_{n}\left(q_{n k} r \right) \, J_{n}\left(q_{n \pm 2, p} r \right) \, d r&{\hbox{(10)}}}$$and Δnk = ∫0R r Jn2 (qnk r) dr, νnk = qnk2+ 2 ∊nk, κnk = − 1 + 6 ∊nk, ∊nk = (1−snk)/4, snk being the value of sq in (4) for q = qnk.

SECTION III

RESULTS AND DISCUSSION

The coupling terms proportional to Formula in (6) are the consequence of volume–charge magnetostatic effects. They are all of the order of (1 − sq ). One has that (1 − sq) ≪ 1 up to q ∼ 1 in ultrathin layers with Formula [see (4)]. If one neglects these terms altogether, one obtains a system of fully decoupled equations for individual cylindrical spin waves, characterized by the dispersion relationFormula TeX Source $$\eqalignno{\omega^2 (q; \cos \theta_0) &= \left(\!q^2 + { 1 - s_q\over 2} \!\right)\! \left(\!q^2 + s_q + { 1 - 3 s_q\over 2} \, \cos^2 \theta_0 \!\right)\qquad&\hbox{(11)}}$$which is obtained from (8) in the limit α → 0. However, the time-periodic coupling terms may give rise to parametric instabilities. Interestingly, these instabilities are governed by a small number of dominant terms, which can be identified by using the asymptotic formula Formula in the equation expressing boundary conditions. One obtains the estimate qnk≃ π (2 s + 1)/4 R, where s = | n| + 2 k. When this approximate expression is used for qn ± 2, p in (10), one obtainsFormula TeX Source $$\eqalignno{n \ge 2 & :\quad \Delta_{n k; p}^{\pm } \simeq \delta_{p, k \mp 1},\quad \Delta_{n 0; p}^{+} \simeq 0,\cr n = \pm 1 & :\quad \Delta_{1k; p}^{-} = \Delta_{-1,k; p}^{+} = \delta_{p k},\cr n \le -2 & :\quad \Delta_{n k; p}^{\pm } \simeq \delta_{p, k \pm 1}, \quad \Delta_{n 0; p}^{-} \simeq 0.&\hbox{(12)}}$$

These relations have an important physical consequence, which is best appreciated by rewriting (5) in the form: δm = ∑s = 0δm(s), whereFormula TeX Source $$\delta {\bf m}^{(s)} = \displaystyle \sum_{\vert n\vert + 2k = s} {\bf {a}}_{n k} (t) \, J_n \left(q_{n k} r \right) \, \exp (i n \phi).\eqno{\hbox{(13)}}$$Under the approximation (12), one finds from (6) that δm(s1) is decoupled from δm(s2) for any s2s1. On the other hand, for each s, the (s+1) cylindrical waves (namely, n = s, s−2, …, −s +2, −s) involved in δm(s) form a 1-D chain, in the sense that only neighboring waves in the aforementioned list are coupled. The absence of coupling between distinct chains would be complete if the approximation qnk≃ π (2 s + 1)/4 R were exact. In this case, all the cylindrical waves in δm(s) would be characterized by exactly the same wave-vector amplitude.

Figure 2
Fig. 2. (a) Magnification of Fig. 1. Labels s = 1,2,3 identify the perturbation chain responsible for the corresponding instability tongue. The pair of dashed lines accompanying each of the s = 2 and s = 3 tongues (one line only for s = 1) represents the parametric resonance condition ω (qnk; cos θ0) = ω0 for the largest and smallest qnk in the chain. Horizontal line at β/α = 0.15 is line along which the computer simulations shown in (b) and (c) were carried out. (b) and (c) Magnitude mavg of average in-plane magnetization obtained from numerical integration of (1) under decreasing (b) and increasing (c) external magnetic field. Dashed line represents the prediction of (2) for sinθ0. Snapshots illustrate magnetization patterns appearing just after the instability jumps. Vertical dotted lines are guides for the eye to compare thresholds with theoretical predictions obtained from (a).

Instabilities are governed by the multipliers of the one-period map [Perko 1996] associated with the dynamics of δm(s). We have used (6) and (12) to make a numerical study of these multipliers for different chains, in order to obtain the instability pattern associated with each of them. The results for a permalloy disk with radius R = 135 nm and thickness d = 3.43 nm are shown in Fig. 1. The band (O+SW) in between the P and A regions is where magnetization precession occurs. Spin-wave instabilities appear in region SW. Each chain δm(s) provides a distinct instability channel. In particular, chains s = 1, s = 2, and s = 3 (s = 0 yields no instability at all) give rise to well-separated instability regions that can be neatly resolved, as shown in Fig. 2(a). In general, the sth chain gives rise to an instability tongue around the parametric resonance condition ω (q; cos θ0) ∼ ω0, where ω (q; cos θ0) is given by (11) and q is of the order of the wave-vector amplitudes involved in the chain [see dashed lines in Fig. 2(a)]. According to parametric resonance theory, resonance occurs for ω = n ω0/2, n = 1, 2, …. The dominant, lowest threshold resonance occurs for n = 1, i.e., at ω = ω0/2. However, one can see from (6) that the parametric frequency is 2 ω0 rather than ω0, which explains why the resonance condition is ω ∼ ω0. This peculiarity is the consequence of the rotational invariance of the problem, and is expected to disappear in situations with broken rotational symmetry. Interestingly, Fig. 1 reveals that, for a given precession amplitude cos θ0 (see dashed line), applying larger fields and currents has a stabilizing effect on the precession. Also, larger fields stabilize precessions of given frequency ω0 = β/α.

To test the predictions of the theory, we have carried out computer simulations based on the numerical integration of (1) by the methods discussed in [d'Aquino 2005]. Simulations were carried out by slowly varying the external magnetic field under constant current. As shown in Fig. 2(b), at large fields the magnitude mavg of the average in-plane magnetization is in full agreement with the prediction of (2) for spatially uniform precession. Then, under decreasing field mavg exhibits well-pronounced jumps, whose positions agree with the theoretical instability thresholds for the s = 2 and s = 3 chains within ten percentage. Beyond these jumps, nonuniform modes appear in the dynamics [see Fig. 2(b)], characterized by twofold and threefold patterns that are consistent with the symmetry of the cylindrical waves involved in the s = 2 and s = 3 chains, respectively. It is worth remarking that the main precession is not completely disrupted, but only somewhat obscured by spin-wave instabilities. Agreement with the theory is also confirmed by the hysteresis in the instability thresholds occurring under decreasing or increasing external field [see Fig. 2(c)]. Future work will be devoted to extending the present approach to more general, nonuniaxial geometries.

Acknowledgment

This work was supported in part by the Ministry of Scientific Research through the Project PRIN-2006098315 (Italy), in part by the European Union (European Social Fund), and in part by the Regione Autonoma Valle d’Aosta (Italy).

Footnotes

Corresponding author: R. Bonin (bonin.roberto@gmail.com).

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G. Bertotti

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R. Bonin

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M. d'Aquino

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C. Serpico

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I. D. Mayergoyz

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